The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper we present the generalization of this result to the ND spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler-Coulomb potential together with N centrifugal terms, is shown to be endowed with (2N-1) functionally independent integrals of the motion: one of them is quartic and the remaining ones are quadratic. The transition from the proper Kepler-Coulomb potential, with its associated quadratic Laplace-Runge-Lenz N-vector, to the generalized system is fully described. The role of spherical, nonlinear (cubic), and coalgebra symmetries in all these systems is highlighted.
Deep Dive into Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces.
The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper we present the generalization of this result to the ND spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler-Coulomb potential together with N centrifugal terms, is shown to be endowed with (2N-1) functionally independent integrals of the motion: one of them is quartic and the remaining ones are quadratic. The transition from the proper Kepler-Coulomb potential, with its associated quadratic Laplace-Runge-Lenz N-vector, to the generalized system is fully described. The role of spherical, nonlinear (cubic), and coalgebra symmetries in all the
The Kepler-Coulomb (KC) potential on Riemannian spaces of constant curvature was already studied by Lipschitz and Killing in the 19th century, and later rediscovered by Schrödinger [1] (see [2] for a detailed discussion). In terms of a geodesic radial distance r between the particle and the origin of the space, the KC potential on the N-dimensional (ND) spherical S N , Euclidean E N and hyperbolic H N spaces reads (see, e.g., [3,4,5] and references therein)
In this paper we shall deal with the integrability properties of the so-called ND generalized KC system, which is defined as the superposition of the (curved) KC potential with N ‘centrifugal’ terms. In the ND Euclidean space E N such a system reads
where K and b i (i = 1, . . . , N) are real constants. This system was known to be quasimaximally superintegrable [3] in the Liouville sense [6], since a set of 2N -2 functionally independent quadratic integrals of motion (including the Hamiltonian) were explicitly known. In fact, in the remarkable classification on superintegrable systems on E 3 by Evans [7], this Hamiltonian was called ‘weakly’ or minimally superintegrable since it had one integral of motion more (four) than the necessary number to be completely integrable (three), but one less than the maximum possible number of independent integrals for a 3D system (five).
In contrast, it was also well known that when at least one of the centrifugal terms vanishes (we shall call this case the quasi-generalized KC system), the resulting Hamiltonian turns out to be maximally superintegrable in arbitrary dimension since a maximal set of 2N -1 functionally independent and quadratic integrals of the motion is explicitly known (see [3,7,8,9,10] and references therein). Moreover, such maximal superintegrability of the quasi-generalized KC system has also been proven for the spherical and hyperbolic spaces [11,12,13] as well as for the Minkowskian and (anti-)de Sitter spacetimes [4,14].
Nevertheless, in a recent work Verrier and Evans [15] have shown that the generalized KC system on E 3 (i.e., the superposition of the 3D KC potential with three centrifugal terms) is maximally suprintegrable, but the additional integral of motion is quartic in the momenta. The aim of this paper is to show that this result holds for an arbitrary dimension N and, moreover, that the generalized KC system is also maximally superintegrable on the ND curved Riemannian spaces of constant curvature: the spherical S N and hyperbolic H N spaces. In this way the ND Euclidean system arises as a smooth limiting flat case that can be interpreted as a contraction in terms of the curvature parameter κ.
In order to prove this result we shall explicitly construct the set of 2N -1 functionally independent integrals of motion for the generalized KC Hamiltonian. In particular, 2N -2 of them will be quadratic and provided by an sl(2, R) Poisson coalgebra symmetry [3,16] (together with the Hamiltonian), while the remaining ‘hidden’ one is quartic in the momenta and generalizes the result of [15] on E 3 to these three ND classical spaces of constant curvature. In this way the short list of ND maximally superintegrable Hamiltonians (see [17] an references therein) is enlarged with another instance.
The paper is organized as follows. In the next section we introduce the geometric background on which the rest of the paper will be based: the Poincaré and Beltrami phase spaces arising, respectively, as the stereographic and the central projection from a linear ambient space R N +1 [3,18]. The next section is devoted to recall the description of the maximal integrability of the curved KC system in terms of these two phase spaces. In section 4 the spherical and ‘hidden’ nonlinear symmetries of the KC system are fully described, thus providing a detailed explanation of the techniques making possible the ’transition’ from the integrability properties of the curved KC system to the generalized one. The core of the paper is contained in section 5, where we explicitly show how the spherical symmetry breaking induced by the centrifugal terms can be appropriately replaced by an sl(2, R) Poisson coalgebra symmetry [19,20] that allows us to construct the ‘additional’ quartic integral of motion for the ND generalized curved KC systems. Finally, some remarks close the paper.
To start with we present the structure of the Poincaré and Beltrami phase spaces [3], which will allow us to deal with the three classical Riemannian spaces in a unified setting. Furthermore, as a new result, we also present the explicit canonical transformation relating both of them.
Given a constant sectional curvature κ, the ND spherical S N (κ > 0), Euclidean E N (κ = 0) and hyperbolic H N (κ < 0) spaces can be simultaneously embedded in an ambient linear space R N +1 with ambient (or Weiertrass) coordinates (x 0 , x) = (x 0 , x 1 , . . . , x N ) by requiring them to fulfill the ‘sphere’ constraint Σ: x 2 0 + κ x 2 = 1. Hereafter for any
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
